Problem 78 - Algebra for College Students

Question

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{x^{2}}=\frac{x+2}{x^{2}}$$

Step-by-Step Solution

Answer

No solution; inconsistent equation.

Step 1: Combine the fractions on the left side

Combine the two fractions on the left side by adding their numerators over the common denominator. equation: \(\frac{1}{x} + \frac{1}{x^{2}} = \frac{x+2}{x^{2}}\) becomes: \(\frac{x + 1}{x^{2}} = \frac{x + 2}{x^{2}}\)

Step 2: Set the numerators equal

Since the denominators are equal, set the numerators equal to each other: \(x + 1 = x + 2\)

Step 3: Solve for x

Subtract \(x\) from both sides of the equation to get: \(1 = 2\). This is a contradiction, so there is no solution.

Step 4: Determine the type of equation

Since the equation leads to a contradiction, it is an inconsistent equation.

Key Concepts

conditional equation

A conditional equation is an equation that is true only under certain conditions or for specific values of the variables involved. Let's take an example: \(2x + 3 = 7\). To solve for \(x\), isolate the variable by performing operations that keep the equation balanced:
\[2x + 3 - 3 = 7 - 3 \] \[2x = 4 \] \[x = 2 \]
Here, \(x = 2\) is the specific condition that makes the equation true. If \(x\) takes any other value, the equation won't hold. Thus, a conditional equation depends on fulfilling certain criteria to be valid.

identity equation

An identity equation is true for all values of the variable(s) involved. This means that no matter what value you substitute for the variable, the equation will hold. For example, consider the equation: \[2(x + 3) = 2x + 6 \]
Let's distribute and simplify both sides:
\[2x + 6 = 2x + 6 \]
The simplified form shows that both sides are always equal, regardless of the value of \(x\). Identity equations provide a foundation for many algebraic principles and proofs, as they represent universal truths in mathematics.

interval notation

Interval notation is a shorthand way of representing a set of numbers along a number line. It's used to describe solution sets, especially in inequalities. Let's break down the components:
\(1.\) **Brackets:** \[ \text{[ ]}\text{ - including the endpoint, e.g., [1, 5]} \] \[ \text{( )}\text{ - excluding the endpoint, e.g., (1, 5)} \]
\(2.\) **Intervals:** \[ \text{Finite Interval - Includes endpoints: [a, b] ; excludes endpoints: (a, b)} \] \[ \text{Infinite Interval - Open-ended: (a, ∞) or (-∞, b]} \]
Interval notation makes it easy to concisely express the range of values that solve an equation or inequality. For example, if \[ x > 3 \], we write this as \[ (3, \infty ) \]. This expresses that \(x\) can be any number greater than 3 but not 3 itself.